Per Martin-Lof. A theory of types. Technical Report 71-3, University of Stockholm, 1971.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is not the only purpose of formalism, and certainly not the predominant one. The predominant one is surely: cut out the crap. 4 Paul Taylor s recent book on mathematical foundations [Tay99] has 33 references to it in its index. 5 For example, Per Martin Lof s first formulation of type theory [ML71] was an inconsistent logic. Still, most proofs people were likely to produce in that theory would happily travel to consistent theories. 5 Mathematicians often share more of their intuition than their use of formal logic would suggest. In a way, formality ensures that you do not start talking ....

Per Martin-Lof. A theory of types. Technical Report 71--3, University of Stockholm, 1971.

....We also need a number of basic data types like the set N of natural numbers and the set List(A) of lists of elements in a set A . However, many specifications can still not be expressed with these sets but require a universe. 3 The need of a universe Martin Lof s first formulation of type theory [10] contained a universe V in which all sets were elements, including V itself. Such a universe would have been very practical to use but, by Girard s paradox, V 2 V implies that all sets are non empty; hence, it is impossible to interpret propositions as sets. In Martin Lof [12] the universe V is ....

Per Martin-Lof. A Theory of Types. Technical Report 71--3, University of Stockholm, 1971.

....derive it, up to possible use of the conversion rule. Since the conversion rule doesn t change the subject between its major premise and its conclusion, the shape of the subject gives no information on when to use this rule. Our technique to remove this ambiguity is due originally to Martin L of [Mar71b] and made known to me by [Hue89] Since the subject of a judgement determines its predicate, at best, only up to conversion, it is clear that the conversion rule must be available at the end of a derivation to fix up the type if necessary. We will try to permute the conversion rule down ....

Per Martin-L of. A theory of types. Technical Report 71-3, University of Stockholm, 1971.

....B(lft(A) B) x) c) la:A. lb:B(a) b) A. B: AType. c: S (A) B) B(lft(A) B) c) Generalizing the way we defined cartesian product in terms of universal types only, we can attempt to define S without using existential types. This plan however fails; here is the best we can do [Martin Lf 71] S = lA. lB:AType. C. a:A.B(a) C) C : A. B:AType. Type pair = lA. lB:AType. la:A. lb:B(a) lC. lc: a:A.B(a) C) c(a) b) A. B:AType. a:A.B(a) S(A) B) unpair = lA. lB:AType. lC. lp:S(A) B) lq: a:A.B(a) C) p(C) q) A. B:AType. C. S(A) B) a:A.B(a) C) C The problem is that ....

P.Martin-Lf, A theory of types, Report 71-3, Dept of Mathematics, University of Stockholm, February 1971, revised October 1971. Page 25

....of this method: a typed and an untyped approach. In the former approach we dene a computability predicate over well typed terms. This technique was developed by Tait (1967) in order to prove normalization for G#del s system T. His method was extended by Girard (1971) for his System F and by Martin L#f (1971) for his type theory. The second approach is similar to Kleene s realizability intepretation (Kleene 1945) but in this approach formulas are realized by (not necessarily well typed) terms rather than G#del numbers. This technique was developed by Tait (1975) where he uses this method to obtain a ....

....A realizability interpretation of Martin L#f s type theory 3 Lemma 2.4 The reduction dened above has the Church Rosser property; that is, if a b and a c then there exists an expression, d, such that b d and c d. The proof of this lemma is similar to the proof of Church Rosser in (MartinL #f 1971). As consequences of the Church Rosser lemma we have the following two corollaries: Corollary 2.5 If an expression has a normal form, then this normal form is unique. Corollary 2.6 The relation = is an equivalence relation. 3 The syntactic model. A problem when dening a realizability ....

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P. Martin-L#f. A Theory of Types. Technical Report 713, University of Stockholm, 1971.

....Constructions The Calculus of Constructions (CC) was introduced by T. Coquand and Huet ( CH88] Coq85] It can be viewed as a unification of Girard s impredicative system F and dependent types, which are the base of Martin Lof s Type Theory. When Martin Lof initially proposed a Type Theory [Mar71] he also attempted to capture Girard s system. However, it turned out that this system was inconsistent because it was possible to encode System U in it. Subsequently Martin Lof avoided this problem by restricting himself to a predicative theory. In a way we may consider CC as a fix to an early ....

Per Martin-Lof. A theory of types. Technical report, University of Stockholm, 1971.

....duplication of work. In these cases just mentioned, we depend on the constructor of the derivation to find the common substructures, but some formal systems duplicate work in such a uniform way that we can give an alternative system that shares some common substructures by construction. Martin Lof [Mar71] gives an algorithm for type synthesis in his impredicative system (now called ) that transforms official derivations to avoid duplicate work. This idea is used in Huet s Constructive Engine [Hue89] an abstract explanation and machine checked proof of correctness of this transformation on ....

Per Martin-Lof. A theory of types. Technical Report 71-3, University of Stockholm, 1971.

....It is well known that we have normalization for Martin L#f s type theory, but it has not actually been shown for the polymorphic type theory with intensional equality and the set of small sets (the rst Universe) Martin L#f has given three normalization proofs. The rst he gave was for the theory [9] where we have a type of all types. This theory was later shown to be inconsistent by Girard. The reason why normalization could be shown was because it was proved in the theory itself extended with a reAEection principle and used the inconsistent axiom U 2 U. The second proof was made for a ....

....expression B satisfying 1 or 2. a. The predicate D Phi (A) holds. b. Phi A is de ned to be Phi B . Some concrete examples of how these computability predicates are de ned will be given in section 3.3. 3.2 Some properties of the computability predicate. Martin L#f s rst version of type theory [9] was based on an impredicative axiom U 2 U which expresses that there is a set of all sets. In the same paper he also gives a normalization proof for the theory. However it was later shown by Girard that this theory was inconsistent. We could try to do something similar by de ning Phi U as iThe ....

Per Martin-L#f. A Theory of Types. Technical Report 713, University of Stockholm, 1971.

....original aims with type theory was that it could serve as a framework in which other theories could be interpreted. And a normalization proof for type theory would then immediately give normalization for a theory expressed in type theory. In Martin L#f s rst formulation of type theory from 1971 [28], theories like rst order arithmetic, G#del s T [18] second order logic and simple type theory [5] could easily be interpreted. However, this formulation contained a reAEection principle expressed by a universe V and including the axiom V2V, which was shown by Girard to be inconsistent. Coquand ....

....type theory [5] could easily be interpreted. However, this formulation contained a reAEection principle expressed by a universe V and including the axiom V2V, which was shown by Girard to be inconsistent. Coquand and Huet s Calculus of Constructions [8] is closely related to the type theory in [28]: instead of having a universe V, they have the two types Prop and Type and the axiom Prop 2 Type, thereby avoiding Girard s paradox. Martin L#f s later formulations of type theory have all been predicative; in particular second order logic and simple type theory cannot be interpreted in them. The ....

Per Martin-L#f. A Theory of Types. Technical Report 713, University of Stockholm, 1971.

....the left premise of Lda) they maintain validity. This is more in keeping with implementations which are actually used, where we work in a current context of mathematical assumptions. We present such a system in table 8, and show it is equivalent to . The idea is originally due to Martin Lof [Mar71] and is used in [Hue89] This system has two judgements, a type judgment of shape Gamma lv M : A (lvtyp) and a validity judgement of shape Gamma lv (lvcxt) Note that they are not mutually inductive: validity depends on typing, but not conversely. We have proved that lv characterizes ....

Per Martin-Lof. A theory of types. Technical Report 71-3, University of Stockholm, 1971.

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